Factoring Quadratic Trinomials5/6/01

Objective: you will factor quadratic polynomials into a pair of binomials

OUTLINEI. Put in descending order of exponents like ax²+bx+c and

multiply a*cII. Find two numbers whose product is (a*c), BUT whose sum is

‘b’ III. Rewrite ax²+ [larger factor]x + [smaller factor]x +c ( larger

factor takes the same sign as ‘b’)IV. Factor the first pair of terms and factor the second pair of

terms(by dividing out the common factor for each pair and if the 3rd term is negative, then factor it out also)

V. Factor out the common binomial factorVI. Write as a pair of binomials

Example 1Factor x²+6x+8

Order and multiply a*c Find factors of (a*c), but

their sum is ‘b’ Write the new polynomial Double Factor Factor the binomial Write a pair of binomials

Factors of 8 Sum of 6? 1,8 9 2,4 6 -1,-8 -9 -2,-4 -6

x²+4x+2x+8 x(x+4) +2(x+4) x(x+4) +2(x+4) (x+4)(x+2)

Example 2Factor x²–4x -21

Factors of –21,Sum of –4? 1,-21 -20 3,-7 -4 -1,21 20 -3,7 4

x² -7x+3x – 21 x(x-7) +3(x-7) x(x-7) +3(x-7) (x-7) (x+3)

•Order and multiply a*c•Find factors of (a*c), but their sum is ‘b’

•Rewrite the new polynomial•Double Factor •Factor the binomial•Write a pair of binomials

Example 3Factor y²+7y-30

Order and multiply a*c Find factors of (a*c), but their

sum is ‘b’

Rewrite the new polynomial Double Factor Factor the binomial Write a pair of binomials

Factor –30 Sum of 7? 1,-30 -29 2,-15 -13 3,-10 -7 5,-6 -1 -1,30 29 -2,15 13 -3, 10 7 -5,6 1

y²+ 10y –3y –30 y(y+10) –3(y+10) y(y+10) –3(y+10) (y+10) (y-3)

Example 4Factor 2x² –13x +15

Order and multiply a*c Find factors of (a*c), but their

sum is ‘b’

Rewrite the new polynomial Double Factor Factor the binomial Write a pair of binomials

Factors of 30 Sum of –13? 1,30 31 2,15 17 3,10 13 5,6 11 -1-,30 -31 -2,-15 -17 -3,- 10 -13 -5,-6 -11

2x² -10x -3x +15 2x(x -5) -3(x -5) 2x(x -5) -3(x -5) (x -5)(2x –3)

Example 5Factor 15r² +44r +21

Order and multiply a*c Find factors of (a*c), but their

sum is ‘b’

Rewrite the new polynomial Double Factor Factor the binomial Write a pair of binomials

Factors of 315 Sum of 44? 1, 315 316 3, 105 108 5, 63 68 7, 45 52 9, 35 44

15r² +35r +9r +21 5r(3r +7) +3(3r +7) 5r(3r +7) +3(3r +7) (3r +7)(5r +3)

Check your understanding(click to check each step)

6) Factor x² +2x –80 7)Factor 3y² -22y –16 x² +2x –80 (no peeking) Factors of –80 Sum of 2? -1, 80 79 -2, 40 38 -4, 20 16 -8, 10 2

x² +10x –8x –80 x(x +10) –8(x +10) x(x +10) –8(x +10) (x +10)(x –8)

• 3y² -22y –16 • Factors of-48 Sum of –22• 1, -48 -47• 2, -24 -22

• 3y² -24y +2y –16• 3y(y –8) +2(y –8)• 3y(y –8) +2(y –8)• (y -8)(3y +2)• Now practice !!!